Absolute neighbourhood retracts and spaces of holomorphic maps from Stein manifolds to Oka manifolds

TL;DR

This paper proves that under certain conditions, the space of holomorphic maps from Stein to Oka manifolds is a deformation retract of the space of continuous maps, using Oka theory and absolute neighbourhood retracts.

Contribution

It establishes that the space of holomorphic maps is a deformation retract of continuous maps for Stein manifolds with specific exhaustions and Oka manifolds, extending previous results.

Findings

The deformation retraction holds for Stein manifolds with finitely many critical points.

The result applies to affine algebraic Stein manifolds.

The proof uses absolute neighbourhood retracts and the parametric Oka property.

Abstract

The basic result of Oka theory, due to Gromov, states that every continuous map $f$ from a Stein manifold $S$ to an elliptic manifold $X$ can be deformed to a holomorphic map. It is natural to ask whether this can be done for all $f$ at once, in a way that depends continuously on $f$ and leaves $f$ fixed if it is holomorphic to begin with. In other words, is $\scrO(S,X)$ a deformation retract of $\scrC(S,X)$? We prove that it is if $S$ has a strictly plurisubharmonic Morse exhaustion with finitely many critical points; in particular, if $S$ is affine algebraic. The only property of $X$ used in the proof is the parametric Oka property with approximation with respect to finite polyhedra, so our theorem holds under the weaker assumption that $X$ is an Oka manifold. Our main tool, apart from Oka theory itself, is the theory of absolute neighbourhood retracts. We also make use of the mixed model structure on the category of topological spaces.